Budarina, Natalia On a problem of Bernik, Kleinbock and Margulis. (English) Zbl 1268.11098 Glasg. Math. J. 53, No. 3, 669-681 (2011). The author considers the system of inequalities \[ | P(x) | < \Psi_1(H(P)), \quad | P'(x) | < \Psi_2(H(P)), \] where \(P\) is an integer polynomial, \(H(P)\) is the maximum among the absolute values of the coefficients of \(P\) and the \(\Psi_i\) are real, positive functions. The set \({\mathcal P}_n(\Psi_1, \Psi_2)\) consists of the numbers \(x \in [-1/2, 1/2]\) for which this system of inequalities is satisfied for infinitely many integer polynomials \(P\) of degree at most \(n\).Under various technical restrictions on the functions \(\Psi_1\) and \(\Psi_2\), the Lebesgue measure of the set \({\mathcal P}_n(\Psi_1, \Psi_2)\) is calculated. With the technical assumptions ignored, the set is null or full according to the convergence or divergence of the series \(\sum_{h=1}^\infty h^{n-2} \Psi_1(h) \Psi_2(h)\), a result very much like the classical Khintchine theorem. The paper is a first step towards establishing a Khintchine type theorem in this context, although there is still a long way to go due to the technical conditions. The proof uses Sprindzhuk’s method of essential and inessential domains along with more recent results. Reviewer: Simon Kristensen (Aarhus) MSC: 11J83 Metric theory 11K60 Diophantine approximation in probabilistic number theory 11J13 Simultaneous homogeneous approximation, linear forms Keywords:Diophantine approximation; approximation by algebraic numbers; metric theory PDF BibTeX XML Cite \textit{N. Budarina}, Glasg. Math. J. 53, No. 3, 669--681 (2011; Zbl 1268.11098) Full Text: DOI References: [1] DOI: 10.1155/S1073792801000241 · Zbl 0986.11053 · doi:10.1155/S1073792801000241 [2] DOI: 10.1112/blms/bdn116 · Zbl 1180.11025 · doi:10.1112/blms/bdn116 [3] Bernik, Dokl. Akad. Nauk 413 pp 151– (2007) [4] DOI: 10.1007/s10986-008-9005-9 · Zbl 1143.11337 · doi:10.1007/s10986-008-9005-9 [5] DOI: 10.4007/annals.2007.166.367 · Zbl 1137.11048 · doi:10.4007/annals.2007.166.367 [6] DOI: 10.4064/aa133-4-6 · Zbl 1222.11096 · doi:10.4064/aa133-4-6 [7] Beresnevich, Mem. Amer. Math. Soc. 179 pp 91– (2006) [8] Bernik, Acta Arith. 53 pp 17– (1989) [9] Beresnevich, A panorama of number theory or the view from Baker’s garden (Zürich, 1999) pp 260– (2002) [10] Bernik, Dokl. Nat. Akad. Nauk Belarusi. 50 pp 30– (2006) [11] DOI: 10.1016/j.jnt.2004.09.007 · Zbl 1078.11050 · doi:10.1016/j.jnt.2004.09.007 [12] Beresnevich, Mosc. Math. J. 2 pp 203– (2002) [13] DOI: 10.4064/aa117-1-4 · Zbl 1201.11078 · doi:10.4064/aa117-1-4 [14] DOI: 10.1112/S0010437X10004860 · Zbl 1206.11091 · doi:10.1112/S0010437X10004860 [15] DOI: 10.1023/A:1015662722298 · Zbl 0997.11053 · doi:10.1023/A:1015662722298 [16] Beresnevich, Acta Arith. 90 pp 97– (1999) [17] DOI: 10.1017/CBO9780511565977 · doi:10.1017/CBO9780511565977 [18] DOI: 10.1007/s00222-006-0509-9 · Zbl 1185.11047 · doi:10.1007/s00222-006-0509-9 [19] DOI: 10.2307/120997 · Zbl 0922.11061 · doi:10.2307/120997 [20] DOI: 10.1007/BF01448437 · JFM 50.0125.01 · doi:10.1007/BF01448437 [21] DOI: 10.1112/S0024610702003137 · Zbl 1020.11049 · doi:10.1112/S0024610702003137 [22] DOI: 10.1017/S0305004110000319 · Zbl 1241.11098 · doi:10.1017/S0305004110000319 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.